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Exponential tractability of linear weighted tensor product problems in the worst-case setting for arbitrary linear functionals

机译：任意线性功能最坏情况下线性加权张量产品问题的指数易易易易易易用性

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摘要

We study the approximation of compact linear operators defined over certain weighted tensor product Hilbert spaces. The information complexity is defined as the minimal number of arbitrary linear functionals needed to obtain an epsilon-approximation for the d-variate problem which is fully determined in terms of the weights and univariate singular values. Exponential tractability means that the information complexity is bounded by a certain function that depends polynomially on d and logarithmically on epsilon(-1). The corresponding unweighted problem was studied in Hickernell et al. (2020) with many negative results for exponential tractability. The product weights studied in the present paper change the situation. Depending on the form of polynomial dependence on d and logarithmic dependence on epsilon(-1), we study exponential strong polynomial, exponential polynomial, exponential quasi-polynomial, and exponential (s, t)-weak tractability with max(s, t) = 1. For all these notions of exponential tractability, we establish necessary and sufficient conditions on weights and univariate singular values for which it is indeed possible to achieve the corresponding notion of exponential tractability. The case of exponential (s, t)-weak tractability with max(s, t) 1 is left for future study. The paper uses some general results obtained in Hickernell et al. (2020) and Kritzer and Woiniakowski (2019). (C) 2020 The Author(s). Published by Elsevier Inc.
机译：我们研究了在某些加权张量产品希尔伯特空间上定义的紧凑线性运算符的近似。信息复杂性被定义为获得D变化问题的epsilon近似所需的任意线性函数的最小数量，这在权重和单次数奇异值方面完全确定。指数途径意味着信息复杂性被某种功能界定，其依赖于DOMOMINY上的D和对数上的epsilon（-1）。在HICKERNELL等人中研究了相应的未加权问题。 （2020）对于指数途径具有许多负面结果。本文研究的产品重量改变了情况。取决于多项式依赖于D和对数依赖性对ε（-1）的依赖性的形式，我们研究指数强大多项式，指数多项式，指数的准多项式和指数（S，T） - 具有最大值的疏忽术（S，T） > = 1.对于指数途径的所有概念，我们对权重和单次奇异值的必要条件确实可以实现相应的指数途径概念。具有最大值的指数（s，t）-weak途径<1的情况留给了未来的研究。本文使用了在HICKERNELL等人中获得的一些一般结果。 （2020）和Kritzer和Woiniakowski（2019年）。 （c）2020提交人。 elsevier公司出版

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